src/HOL/Euclidean_Division.thy
 changeset 64785 ae0bbc8e45ad child 66798 39bb2462e681
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Euclidean_Division.thy	Wed Jan 04 21:28:29 2017 +0100
1.3 @@ -0,0 +1,287 @@
1.4 +(*  Title:      HOL/Euclidean_Division.thy
1.5 +    Author:     Manuel Eberl, TU Muenchen
1.6 +    Author:     Florian Haftmann, TU Muenchen
1.7 +*)
1.8 +
1.9 +section \<open>Uniquely determined division in euclidean (semi)rings\<close>
1.10 +
1.11 +theory Euclidean_Division
1.12 +  imports Nat_Transfer
1.13 +begin
1.14 +
1.15 +subsection \<open>Quotient and remainder in integral domains\<close>
1.16 +
1.17 +class semidom_modulo = algebraic_semidom + semiring_modulo
1.18 +begin
1.19 +
1.20 +lemma mod_0 [simp]: "0 mod a = 0"
1.21 +  using div_mult_mod_eq [of 0 a] by simp
1.22 +
1.23 +lemma mod_by_0 [simp]: "a mod 0 = a"
1.24 +  using div_mult_mod_eq [of a 0] by simp
1.25 +
1.26 +lemma mod_by_1 [simp]:
1.27 +  "a mod 1 = 0"
1.28 +proof -
1.29 +  from div_mult_mod_eq [of a one] div_by_1 have "a + a mod 1 = a" by simp
1.30 +  then have "a + a mod 1 = a + 0" by simp
1.31 +  then show ?thesis by (rule add_left_imp_eq)
1.32 +qed
1.33 +
1.34 +lemma mod_self [simp]:
1.35 +  "a mod a = 0"
1.36 +  using div_mult_mod_eq [of a a] by simp
1.37 +
1.38 +lemma dvd_imp_mod_0 [simp]:
1.39 +  assumes "a dvd b"
1.40 +  shows "b mod a = 0"
1.41 +  using assms minus_div_mult_eq_mod [of b a] by simp
1.42 +
1.43 +lemma mod_0_imp_dvd:
1.44 +  assumes "a mod b = 0"
1.45 +  shows   "b dvd a"
1.46 +proof -
1.47 +  have "b dvd ((a div b) * b)" by simp
1.48 +  also have "(a div b) * b = a"
1.49 +    using div_mult_mod_eq [of a b] by (simp add: assms)
1.50 +  finally show ?thesis .
1.51 +qed
1.52 +
1.53 +lemma mod_eq_0_iff_dvd:
1.54 +  "a mod b = 0 \<longleftrightarrow> b dvd a"
1.55 +  by (auto intro: mod_0_imp_dvd)
1.56 +
1.57 +lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]:
1.58 +  "a dvd b \<longleftrightarrow> b mod a = 0"
1.59 +  by (simp add: mod_eq_0_iff_dvd)
1.60 +
1.61 +lemma dvd_mod_iff:
1.62 +  assumes "c dvd b"
1.63 +  shows "c dvd a mod b \<longleftrightarrow> c dvd a"
1.64 +proof -
1.65 +  from assms have "(c dvd a mod b) \<longleftrightarrow> (c dvd ((a div b) * b + a mod b))"
1.67 +  also have "(a div b) * b + a mod b = a"
1.68 +    using div_mult_mod_eq [of a b] by simp
1.69 +  finally show ?thesis .
1.70 +qed
1.71 +
1.72 +lemma dvd_mod_imp_dvd:
1.73 +  assumes "c dvd a mod b" and "c dvd b"
1.74 +  shows "c dvd a"
1.75 +  using assms dvd_mod_iff [of c b a] by simp
1.76 +
1.77 +end
1.78 +
1.79 +class idom_modulo = idom + semidom_modulo
1.80 +begin
1.81 +
1.82 +subclass idom_divide ..
1.83 +
1.84 +lemma div_diff [simp]:
1.85 +  "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> (a - b) div c = a div c - b div c"
1.86 +  using div_add [of _  _ "- b"] by (simp add: dvd_neg_div)
1.87 +
1.88 +end
1.89 +
1.90 +
1.91 +subsection \<open>Euclidean (semi)rings with explicit division and remainder\<close>
1.92 +
1.93 +class euclidean_semiring = semidom_modulo + normalization_semidom +
1.94 +  fixes euclidean_size :: "'a \<Rightarrow> nat"
1.95 +  assumes size_0 [simp]: "euclidean_size 0 = 0"
1.96 +  assumes mod_size_less:
1.97 +    "b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
1.98 +  assumes size_mult_mono:
1.99 +    "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
1.100 +begin
1.101 +
1.102 +lemma size_mult_mono': "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (b * a)"
1.103 +  by (subst mult.commute) (rule size_mult_mono)
1.104 +
1.105 +lemma euclidean_size_normalize [simp]:
1.106 +  "euclidean_size (normalize a) = euclidean_size a"
1.107 +proof (cases "a = 0")
1.108 +  case True
1.109 +  then show ?thesis
1.110 +    by simp
1.111 +next
1.112 +  case [simp]: False
1.113 +  have "euclidean_size (normalize a) \<le> euclidean_size (normalize a * unit_factor a)"
1.114 +    by (rule size_mult_mono) simp
1.115 +  moreover have "euclidean_size a \<le> euclidean_size (a * (1 div unit_factor a))"
1.116 +    by (rule size_mult_mono) simp
1.117 +  ultimately show ?thesis
1.118 +    by simp
1.119 +qed
1.120 +
1.121 +lemma dvd_euclidean_size_eq_imp_dvd:
1.122 +  assumes "a \<noteq> 0" and "euclidean_size a = euclidean_size b"
1.123 +    and "b dvd a"
1.124 +  shows "a dvd b"
1.125 +proof (rule ccontr)
1.126 +  assume "\<not> a dvd b"
1.127 +  hence "b mod a \<noteq> 0" using mod_0_imp_dvd [of b a] by blast
1.128 +  then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
1.129 +  from \<open>b dvd a\<close> have "b dvd b mod a" by (simp add: dvd_mod_iff)
1.130 +  then obtain c where "b mod a = b * c" unfolding dvd_def by blast
1.131 +    with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
1.132 +  with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
1.133 +    using size_mult_mono by force
1.134 +  moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
1.135 +  have "euclidean_size (b mod a) < euclidean_size a"
1.136 +    using mod_size_less by blast
1.137 +  ultimately show False using \<open>euclidean_size a = euclidean_size b\<close>
1.138 +    by simp
1.139 +qed
1.140 +
1.141 +lemma euclidean_size_times_unit:
1.142 +  assumes "is_unit a"
1.143 +  shows   "euclidean_size (a * b) = euclidean_size b"
1.144 +proof (rule antisym)
1.145 +  from assms have [simp]: "a \<noteq> 0" by auto
1.146 +  thus "euclidean_size (a * b) \<ge> euclidean_size b" by (rule size_mult_mono')
1.147 +  from assms have "is_unit (1 div a)" by simp
1.148 +  hence "1 div a \<noteq> 0" by (intro notI) simp_all
1.149 +  hence "euclidean_size (a * b) \<le> euclidean_size ((1 div a) * (a * b))"
1.150 +    by (rule size_mult_mono')
1.151 +  also from assms have "(1 div a) * (a * b) = b"
1.152 +    by (simp add: algebra_simps unit_div_mult_swap)
1.153 +  finally show "euclidean_size (a * b) \<le> euclidean_size b" .
1.154 +qed
1.155 +
1.156 +lemma euclidean_size_unit:
1.157 +  "is_unit a \<Longrightarrow> euclidean_size a = euclidean_size 1"
1.158 +  using euclidean_size_times_unit [of a 1] by simp
1.159 +
1.160 +lemma unit_iff_euclidean_size:
1.161 +  "is_unit a \<longleftrightarrow> euclidean_size a = euclidean_size 1 \<and> a \<noteq> 0"
1.162 +proof safe
1.163 +  assume A: "a \<noteq> 0" and B: "euclidean_size a = euclidean_size 1"
1.164 +  show "is_unit a"
1.165 +    by (rule dvd_euclidean_size_eq_imp_dvd [OF A B]) simp_all
1.166 +qed (auto intro: euclidean_size_unit)
1.167 +
1.168 +lemma euclidean_size_times_nonunit:
1.169 +  assumes "a \<noteq> 0" "b \<noteq> 0" "\<not> is_unit a"
1.170 +  shows   "euclidean_size b < euclidean_size (a * b)"
1.171 +proof (rule ccontr)
1.172 +  assume "\<not>euclidean_size b < euclidean_size (a * b)"
1.173 +  with size_mult_mono'[OF assms(1), of b]
1.174 +    have eq: "euclidean_size (a * b) = euclidean_size b" by simp
1.175 +  have "a * b dvd b"
1.176 +    by (rule dvd_euclidean_size_eq_imp_dvd [OF _ eq]) (insert assms, simp_all)
1.177 +  hence "a * b dvd 1 * b" by simp
1.178 +  with \<open>b \<noteq> 0\<close> have "is_unit a" by (subst (asm) dvd_times_right_cancel_iff)
1.179 +  with assms(3) show False by contradiction
1.180 +qed
1.181 +
1.182 +lemma dvd_imp_size_le:
1.183 +  assumes "a dvd b" "b \<noteq> 0"
1.184 +  shows   "euclidean_size a \<le> euclidean_size b"
1.185 +  using assms by (auto elim!: dvdE simp: size_mult_mono)
1.186 +
1.187 +lemma dvd_proper_imp_size_less:
1.188 +  assumes "a dvd b" "\<not> b dvd a" "b \<noteq> 0"
1.189 +  shows   "euclidean_size a < euclidean_size b"
1.190 +proof -
1.191 +  from assms(1) obtain c where "b = a * c" by (erule dvdE)
1.192 +  hence z: "b = c * a" by (simp add: mult.commute)
1.193 +  from z assms have "\<not>is_unit c" by (auto simp: mult.commute mult_unit_dvd_iff)
1.194 +  with z assms show ?thesis
1.195 +    by (auto intro!: euclidean_size_times_nonunit)
1.196 +qed
1.197 +
1.198 +end
1.199 +
1.200 +class euclidean_ring = idom_modulo + euclidean_semiring
1.201 +
1.202 +
1.203 +subsection \<open>Uniquely determined division\<close>
1.204 +
1.205 +class unique_euclidean_semiring = euclidean_semiring +
1.206 +  fixes uniqueness_constraint :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
1.207 +  assumes size_mono_mult:
1.208 +    "b \<noteq> 0 \<Longrightarrow> euclidean_size a < euclidean_size c
1.209 +      \<Longrightarrow> euclidean_size (a * b) < euclidean_size (c * b)"
1.210 +    -- \<open>FIXME justify\<close>
1.211 +  assumes uniqueness_constraint_mono_mult:
1.212 +    "uniqueness_constraint a b \<Longrightarrow> uniqueness_constraint (a * c) (b * c)"
1.213 +  assumes uniqueness_constraint_mod:
1.214 +    "b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> uniqueness_constraint (a mod b) b"
1.215 +  assumes div_bounded:
1.216 +    "b \<noteq> 0 \<Longrightarrow> uniqueness_constraint r b
1.217 +    \<Longrightarrow> euclidean_size r < euclidean_size b
1.218 +    \<Longrightarrow> (q * b + r) div b = q"
1.219 +begin
1.220 +
1.221 +lemma divmod_cases [case_names divides remainder by0]:
1.222 +  obtains
1.223 +    (divides) q where "b \<noteq> 0"
1.224 +      and "a div b = q"
1.225 +      and "a mod b = 0"
1.226 +      and "a = q * b"
1.227 +  | (remainder) q r where "b \<noteq> 0" and "r \<noteq> 0"
1.228 +      and "uniqueness_constraint r b"
1.229 +      and "euclidean_size r < euclidean_size b"
1.230 +      and "a div b = q"
1.231 +      and "a mod b = r"
1.232 +      and "a = q * b + r"
1.233 +  | (by0) "b = 0"
1.234 +proof (cases "b = 0")
1.235 +  case True
1.236 +  then show thesis
1.237 +  by (rule by0)
1.238 +next
1.239 +  case False
1.240 +  show thesis
1.241 +  proof (cases "b dvd a")
1.242 +    case True
1.243 +    then obtain q where "a = b * q" ..
1.244 +    with \<open>b \<noteq> 0\<close> divides
1.245 +    show thesis
1.246 +      by (simp add: ac_simps)
1.247 +  next
1.248 +    case False
1.249 +    then have "a mod b \<noteq> 0"
1.250 +      by (simp add: mod_eq_0_iff_dvd)
1.251 +    moreover from \<open>b \<noteq> 0\<close> \<open>\<not> b dvd a\<close> have "uniqueness_constraint (a mod b) b"
1.252 +      by (rule uniqueness_constraint_mod)
1.253 +    moreover have "euclidean_size (a mod b) < euclidean_size b"
1.254 +      using \<open>b \<noteq> 0\<close> by (rule mod_size_less)
1.255 +    moreover have "a = a div b * b + a mod b"
1.256 +      by (simp add: div_mult_mod_eq)
1.257 +    ultimately show thesis
1.258 +      using \<open>b \<noteq> 0\<close> by (blast intro: remainder)
1.259 +  qed
1.260 +qed
1.261 +
1.262 +lemma div_eqI:
1.263 +  "a div b = q" if "b \<noteq> 0" "uniqueness_constraint r b"
1.264 +    "euclidean_size r < euclidean_size b" "q * b + r = a"
1.265 +proof -
1.266 +  from that have "(q * b + r) div b = q"
1.267 +    by (auto intro: div_bounded)
1.268 +  with that show ?thesis
1.269 +    by simp
1.270 +qed
1.271 +
1.272 +lemma mod_eqI:
1.273 +  "a mod b = r" if "b \<noteq> 0" "uniqueness_constraint r b"
1.274 +    "euclidean_size r < euclidean_size b" "q * b + r = a"
1.275 +proof -
1.276 +  from that have "a div b = q"
1.277 +    by (rule div_eqI)
1.278 +  moreover have "a div b * b + a mod b = a"
1.279 +    by (fact div_mult_mod_eq)
1.280 +  ultimately have "a div b * b + a mod b = a div b * b + r"
1.281 +    using \<open>q * b + r = a\<close> by simp
1.282 +  then show ?thesis
1.283 +    by simp
1.284 +qed
1.285 +
1.286 +end
1.287 +
1.288 +class unique_euclidean_ring = euclidean_ring + unique_euclidean_semiring
1.289 +
1.290 +end
```